%I #10 Jun 17 2022 15:57:51

%S 1,2,6,32,318,5552,159126,7137272,484656318,48628712192,7076367228486,

%T 1471524821492552,432066672598422318,177354805872559516112,

%U 100928502119652298356726,79062670900333522721886872,84733519638342583432646258718,123582326772837258238596562116512,244150974458417420635453430918487846

%N Number of binary relations on [n] that are idempotent and reduced.

%C The Boolean matrix representing a binary relation on [n] is row (column) reduced if no nonzero row (column) is the sum of other rows (columns). It is reduced if it is both row reduced and column reduced.

%C a(n) is the number of partial order relations on Y, where Y is some subset of [n].

%H R. J. Plemmons and M. T. West, <a href="http://dx.doi.org/10.2140/pjm.1970.35.743">On the semigroup of binary relations</a>, Pacific Journal of Mathematics, vol 35, No. 3, 1970. Theorem 2.4

%F E.g.f.: A(x)*exp(x) where A(x) is the e.g.f. for A001035.

%F a(n) = Sum_{k=0..n} binomial(n,k)*A001035(n-k).

%t nn = 18; A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt",

%t "Table"], {_, _}][[All, 2]];A[x_] = Sum[A001035[[n + 1]] x^n/n!, {n, 0, nn}];

%t Range[0, nn]! CoefficientList[Series[A[x] Exp[x], {x, 0, nn}], x]

%Y Cf. A001035, A121337.

%K nonn

%O 0,2

%A _Geoffrey Critzer_, Jun 08 2022